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arXiv · 1501.00887

Hypersymplectic structures with torsion on Lie algebroids

P. Antunes·J. M. Nunes da Costa
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Abstract

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one correspondence between hypersymplectic structures with torsion and hyperkähler structures with torsion. We show that given a Lie algebroid with a hypersymplectic structure with torsion, the deformation of the Lie algebroid structure by any of the transition morphisms does not affect the hypersymplectic structure with torsion. We also show that if a triplet of $2$-forms is a hypersymplectic structure with torsion on a Lie algebroid $A$, then the triplet of the inverse bivectors is a hypersymplectic structure with torsion for a certain Lie algebroid structure on the dual $A^*$, and conversely. Examples of hypersymplectic structures with torsion are included.

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P. Antunes, J. M. Nunes da Costa. 2016-01-26. Hypersymplectic structures with torsion on Lie algebroids. https://doi.org/10.1016/j.geomphys.2016.01.010

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